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I’ve raised this elsewhere, but, despite a lack of encouragement, for some reason the thought keeps pressing itself on me. Sorry if it’s a bad one.
If obstruction to extension is represented by this diagram,
$\array{ \mathbf{B}Q &\stackrel{f}{\to}& \mathbf{B}G &\stackrel{\mathbf{c}}{\to}& \mathbf{B}^n A \\ \downarrow && \downarrow^{\mathrlap{\phi}} & \nearrow_{\mathrlap{\hat \mathbf{c}}} \\ * &\to& \mathbf{B}H }$can’t we see a resemblance to Kan extensions. Whereas in the former case we’re working in a $(\infty, 1)$-setting, with Kan extensions we’re in a 2-categorical setting. Isn’t it that that extra directed dimension allows you to provide best comparative approximations on either side in the shape of right and left extensions? Can we not have something cofiber-like going on in a 2-category?
Or maybe the best place to look would be in an $(\infty, 2)$-setting. Perhaps the cohomological equivalent of that setting relates to the goodness of the approximations of Kan extensions.
One setup for nonabelian homological algebra (which as a special case, may be an alternative way to deal with extensions and higher extensions in much more general setup) is to define satellites via Kan extensions. See
Maybe I’d need to learn about category cohomology.
I guess I can see a “resemblance” in that both concepts are talking about some sort of extension, but I can’t see any precise relationship.
One thing to note is that I think it can happen that a functor $f$ admits an honest extension along a functor $j$ (i.e. a functor $g$ with $g \circ j \cong f$), but neither the right nor the left Kan extension is an honest one (i.e. the canonical transformations $f \to Lan_j f \circ j$ and $Ran_j f \circ j \to f$ are not isomorphisms). At least, at the moment I don’t see any reason why not. If this is the case, then the Kan extension doesn’t measure the “obstruction to existence of an extension” in any obvious sense.
On the other hand, one could consider the non-invertibility of the canonical transformations $f \to Lan_j f \circ j$ and $Ran_j f \circ j \to f$ as themselves being some sort of “obstruction” to having a universal honest extension. But I don’t immediately see any way to regard that as a kind of cohomology.
David,
usually, the cohomology of a category is simply defined (more or less explicitly) to be that of the $\infty$-groupoid that it presents, i.e. of its nerve.
Concerning Kan extensions: true, these are the best left/right approximation to a possibly non-existent extension in a context of “directed homotopy theory”. There are attempts to put this to use in the context of “TQFTs as directed cocycles” in the sense that we discussed recently on the blog. But I cannot really offer a coherent picture, alas.
I wonder what’s even thinkable here. We have fibration in a 2-category. (There are even fibrations in tricategories.) Would there then be fiber sequences in a 2-category, or $(\infty, 2)$-category?
I see p. 13 of Lifting Problems and Transgression for Non-Abelian Gerbes has
We want to prove that this sequence is a “fibre sequence” in bicategories.
By the way, are ’fibration sequence’ and ’fiber sequence’ used interchangeably? At fiber sequence, it begins
A fibration sequence is…
By the way, are ’fibration sequence’ and ’fiber sequence’ used interchangeably?
No guarantee that I have always used it in the following way consistently, but I guess the following is what one should do:
fiber sequence or, for emphasis, homotopy fiber sequence is the abstract notion, indicating that a morphism is the homotopy fiber / $\infty$-fiber of the next.
If we have a model structure around then it is a fact that the homotopy fiber of a morphism is the ordinary fiber of a fibration representative of the morphism. So often in the literature people stick to this presentation of the general situation, demand the morphism on the right to be a fibration and the one on the left to be its ordinary fiber and then speak of a fibration sequence.
So I should go to the entry and change “fibration sequence” to “fiber sequence”.
I wonder what’s even thinkable here.
I always liked to think of the Grothendieck construction of some $F : C \to Cat$ as the lax fiber of $F$. It very much plays the role of a “directed homotopy fiber”, I’d think.
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